72
This Problem is of much use in forming the curvature or easing in the angles for the rail and stairs,
Problem 7.
To find a right line which is nearest equal to the semi-circumference of a circle, as c d the
tangent is to the semicircle a b, Jig. 15.
Let a b be the diameter, having a b as centres, describe the arcs $/and b g, and at the bisecting point of gf, at e, draw straight lines from the point e through a and b to c and d; then draw the tangent line c d parallel to the diameter or base a b: then c d will be nearest equal to the semicircle part, a b. To obtain the lengths of the several segments a h, h i and i b upon the tangent line c d} from the point c draw lines through h and i to the tangent k I; then c 1^ kl,ld, upon the tangent, will be equal to a h, h i and i b of the several segments. .By this process the student will perceive that any portion of the circle may be transferred to a straight line.
Problem 8. To form an ellipsis by means of segments of circles, as fig. 19.
Let a b be the transverse, and c d the conjugate axis, having d as centre, with d c and d h as radius, describe the quadrant c h; then from the centre d draw the radii d s at an acute angle of 69 degrees from d h, the base; then divide the conjugate axis c d into five equal parts; then through the point 1, draw ef parallel to the transverse axis a b; then form an equilateral triangle, as d e /; then having / as centre with / s and/'y as radii, describe the quadrant sv; then from the centre / draw ft at an angle of 53 degrees from d h, the base; then take the distance d 1, and apply said distance from / to i upon the line /1; then having/ as centre, withy t and/ x as radii, describe the quadrant t x; then from, the centre/ draw/ u at an angle of 30 degrees from d h, the base; then at the point of intersection with/ u to the transverse axis, a 5, will be the centre n; then having n as centre, with n u and n w as radii, describe the quadrant u w: then the segment c s of the quadrant d c h, and the segment s t of the quadrant/s v, and the segment t u of the quadrant/1 x, and the segment u b of the